What would be an example that rank of a subgroup of a free group is greater than the rank of free group?

Question

Let $G$ be a free group and $H$ be a subgroup of $G$.

Then, $H$ is also a free group.

Let $R_G,R_H$ be ranks of $G,H$ respectively.

what would be an example that $R_H>R_G$?

Answer 1

I think I have an example, but I can't prove it.

Let $G$ be the free group generated by two elements $a$ and $b$, and $H$ be its subgroup consisting of all elements which are a product of an even number of elements among $a$, $b$, $a^{-1}$ and $b^{-1}$.

Then I'm pretty sure that $R_H = 3$.

Answer 2

Let $G$ be the free group on two elements $a,b$. Let $H$ be the group generated by $g_n = a^nba^{-n}$. H is countably infinitely generated. To get a handle on these arguments the nicest way is to look at covering maps of graphs, and study the fundamental groups (I realize this may be a bit beyond what you'd like to use).